A 11 Re A A01

for all complex A and a real constant A0. The solutions to the problems (6.1.18) and (6.1.23) can be written under this condition in the form m = mm,

C(t) = T(t - t')B(t') dt', Jo where {T(t)},>0 is a family of continuous (over t) operators satisfying the condition

117X011 <pexp(A0i);

fi is a constant.

In the general case, the operators A and B depend on t, and the solutions to the problems (6.1.17), (6.1.18) and (6.1.23) are determined by the evolution operator V:

The solution of Equation (6.1.1) is reduced, with the help of the evolution operator, to the solution of an integral equation for the function u(t) = y(t, a) - y(t, a), i.e.,

u(t) = V{t, 0)u(0) + V(t, t')f(u, t') dt', Jo f(u, a) = v(y, a) - v(y, ä) - A(t, u).

If the operator A is unbounded, it is common for the operator V to have smoothing properties. This allows us to justify the relations (6.1.20)-(6.1.22) for a broad class of problems.

Sensitivity of steady-state and time-periodic solutions. The steady-state solution of Equation (6.1.1) complies with Equation (6.1.2) for G = v. As applied to this case, Equations (6.1.3), (6.1.5), (6.1.8), (6.1.5') and (6.1.9') are written as y(a) - y(ct) = Ç(a - a) + o(||a - â||), AC + B = 0,

R(y, a) - R(y, â) = VR + o(||a - a||), VR = (C*,B( a-a)), A*Ç* = -ry,

where the operators A and B are defined by the formulae (6.1.19), and the functional R satisfies the condition (6.1.6).

The implicit function theorem guarantees correctness of the formulae (6.1.28) if the spectrum of the operator A has no eigenvalues at the origin of the imaginary axis. Violation of this condition is usually accompanied by the loss of stability of equilibrium solutions to small perturbations in the initial data. In general, if the spectrum of the operator A lies strictly in the left-half plane, the formulae (6.1.28) determine a linear change in sensitivity to parameter variations.

We consider the equations with time-periodic coefficients

Let y(t) be a time-periodic solution of Equation (6.1.2) for a fixed value a of the parameter a, i.e.

Equations for variations referred to the problem (6.1.29) are given, as before, by the relations (6.1.17) and (6.1.18) but now the operators A and B are t-periodic functions, and the evolution operator of Equation (6.1.18) satisfies the condition

The operator V(t) = V(r, 0) is known as the monodromy operator, and its eigenvalues are called multiplicators. Suppose that £ = exp(Xt)W(t) in Equation

(6.1.17). If W has a period t, Equation (6.1.17) guarantees the fulfilment of the relations

— = A(t)W(t) + XW(t); W{t + t) = W(t). (6.1.32) di

A set of X for which the problem (6.1.32) has a solution is named as the stability spectrum of the time-periodic solution y(t). In regular cases, multipli-cators coincide with points exp{/lf}. If the monodromy operator has a logarithmic singularity, the standard form of the implicit function theorem can no longer be applied and the Floquet representation V(t, 0) = Q(t) exp(t, T) is valid. Here Q is a i-periodic operator: r = t~ 1 In V(z). The Floquet representation occurs particularly if t"1 jo M(0ll di does not exceed a constant c depending on the structure of the phase space Y. For Hilbert space this constant is equal to n. The Floquet representation allows us to reduce Equations (6.1.17) and (6.1.18) to analogous equations with constant coefficients.

Let us consider the problem of evaluation of the sensitivity of a time-periodic solution assuming that the spectrum of the monodromy operator is strictly inside the unit circle (at a fixed distance from the unit circle). In this case the stability spectrum lies in the left-half plane and the estimate || V(t, i')ll < H exp{z(i - 0} is fulfilled for x < 0.

As applied to time-periodic solutions, the basic equation of sensitivity takes the form y(t, a) — y(t, a) = £(a — a) + o(||a — a||), (6.1.33a)

da J

-dC*/dt = A*(t)t;* - ry; ç*(i + t) = ç*(i), (6.1.33d)

with operators A and B being defined by the formulae (6.1.19) and the functional R satisfying the relations (6.1.21) at T = x.

We derive formulae to evaluate the functions £ and Ç*. Any solution of Equation (6.1.18) admits the representation

Considering that this solution has to be time-periodic it is necessary and

Sensitivity of the climatic system sufficient to have the following equality fulfilled:

Here I is the identical operator. With the indicated assumptions the operator I — V(t) is invertible. Hence,

On the basis of this relation and the formula (6.1.34) we obtain m = nt,o)[i-mr1

The function C* is constructed in a similar way with the help of the operator V*(t, t').

The above results are valid a fortiori if the inequality

with x < 0 is fulfilled. The results can be extended to a broad class of unbounded non-linear problems. However, the assumptions used are violated for auto-oscillations (time-periodic solutions of the autonomous equation (6.1.1)). In such a situation V(z)v(y,a) = v(y, a) so that one multiplicator (equal to 1) is necessarily sited on the unit circle. If the remaining part of the spectrum of the monodromy operator lies strictly inside the unit circle, auto-oscillations are stable and their sensitivities are linear in nature.

Sensitivity at large time intervals. The basic relations (6.1.20)-(6.1.22) of the theory of sensitivity of time-dependent problems are valid under sufficiently general assumptions at any finite time interval [0, T~\, although the accuracy of these relations can diminish exponentially as T increases. It may be possible to ensure that the exponential growth will be absent when a solution under investigation converges to an equilibrium or time-periodic state, the stability spectrum of which lies strictly in the left-half plane. In such a situation the evolution operator of equations of sensitivity satisfies the condition || F(t, i')|| < jj, exp{x(t — t')} with x < 0- One may prove using this condition that solutions of direct and adjoint equations of sensitivity converge exponentially to their limiting values. Accordingly, sensitivity of a limiting state may be calculated by the stabilization method.

The systems, where small perturbations increase exponentially with time, can be examined by probability methods. See, e.g., Kagan et al. (1990) for a more detailed discussion of this aspect of the subject.

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